Ando-Hiai and Golden-Thompson inequalities

Our main target in this article is to present complementary inequalities of Golden-Thompson and Ando-Hiai types. For example, under mild conditions on the positive definite matrices A, B, we show the complementary Golden-Thompson inequality parallel to(e(pA) sigma(v)e(pB))(1/p)parallel to <= theta(1/p) parallel to e(A del v B) parallel to, 0 <= v <= 1, for any matrix mean sigma(v) between the weighted harmonic mean !(v) and the weighted arithmetic mean del v, where parallel to center dot parallel to is an arbitrary unitarily invariant norm and theta being a constant depending on p > 1 and v. Further inequalities of Golden-Thompson and Ando-Hiai types will be presented for the domain v is not an element of [0, 1] and for means other than the geometric mean.